In this paper, a new upper bound on the spectral radius of the Hadamard product of two nonnegative matrices and a new lower bound on the smallest eigenvalue of the Fan product of two nonsingular M-matrices are given, which further improve the results of [5] in some cases. Finally, some examples are given to show that the bounds are better than some previous results.

nonnegative matrices, Hadamard product, Fan product, eigenvalue, M-matrix.